Group extension

In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence

If G is an extension of Q by N, then G is a group, N is a normal subgroup of G and the quotient group G/N is isomorphic to the group Q. Group extensions arise in the context of the extension problem, where the groups Q and N are known and the properties of G are to be determined. Note that the phrasing "G is an extension of N by Q" is also used by some.

Since any finite group G possesses a maximal normal subgroup N with simple factor group G/N, all finite groups may be constructed as a series of extensions with finite simple groups. This fact was a motivation for completing the classification of finite simple groups.

An extension is called a central extension if the subgroup N lies in the center of G.

One extension, the direct product, is immediately obvious. If one requires G and Q to be abelian groups, then the set of isomorphism classes of extensions of Q by a given (abelian) group N is in fact a group, which is isomorphic to

cf. the Ext functor. Several other general classes of extensions are known but no theory exists which treats all the possible extensions at one time. Group extension is usually described as a hard problem; it is termed the extension problem.

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